### Spring 2022

####
**
March 29, 2022 Tue, 3:00-4:00PM
**

Ao Sun, UChicago

**Dynamics of Singularities of Mean Curvature Flow:
**
This is a survey talk on the relations between dynamics and singularity structure of geometric flows. I will use
mean curvature flow as the model to discuss how the dynamics of the singularities would influence the singularity
structure of geometric flows, and how it influences the regularity of the associated level set flow. I will also
discuss the generic behavior of the mean curvature flow, and why generically the singular behavior of the mean
curvature flow is simpler. If time permits, I will discuss our recent work on generic mean curvature flow, including
showing that generically the regularity of the level set flow is low. This is joint work with Jinxin Xue (Tsinghua
University).

####
**
April 5, 2022 Tue, 3:00-4:00PM
**

Vanderson Lima, UFRGS

**TBD:
**
TBD.

####
**
April 12, 2022 Tue, 3:00-4:00PM
**

Zhihan Wang, Princeton

**Generic Regularity for All Minimal Hypersurfaces in 8-Manifolds:
**
The well-known Simons cone suggests that singularities may exist in a stable minimal hypersurface in Riemannian
manifolds of dimension greater than 7, locally modeled on stable minimal hypercones. It was conjectured that
generically they can be perturbed away. In this talk, we present a way to eliminate these singularities by
perturbing metric in an 8-manifold. By combining with a Sard-Type Theorem for space of singular minimal
hypersurfaces of dimension $7$, we prove that in an 8-manifold with generic metric, every locally stable minimal
hypersurface has no singularity. In particular, this proves the existence of infinitely many SMOOTH minimal
hypersurfaces in a generic 8-manifold. This is based on the joint work with Yangyang Li.

####
**
April 19, 2022 Tue, 3:00-4:00PM
**

Peter McGrath, NCSU

**TBD:
**
TBD.

####
**
April 26, 2022 Tue, 3:00-4:00PM
**

Yuchin Sun, University of California Santa Cruz

**TBD:
**
TBD.

####
**
May 3, 2022 Tue, 3:00-4:00PM
**

Alessandro Pigati, NYU Courant

**TBD:
**
TBD.

####
**
May 10, 2022 Tue, 3:00-4:00PM
**

Hongyi Sheng, UC Irvine

**Deformations of the Scalar Curvature and the Mean Curvature:
**
In Riemannian manifold (M^n, g), it is well-known that its minimizing hypersurface is smooth when n<=7, and singular
when n>=8. This is one of the major difficulties in generalizing many interesting results to higher dimensions,
including the Riemannian Penrose inequality. In particular, in dimension 8, the minimizing hypersurface has
isolated singularities, and Nathan Smale constructed a local perturbation process to smooth out the singularities.
However, Smale's perturbation will also produce a small region with possibly negative scalar curvature. In order
to apply this perturbation in general relativity, we constructed a local deformation prescribing the scalar
curvature and the mean curvature simultaneously. In this talk, we will discuss how the weighted function spaces
help us localize the deformation in complete manifolds with boundary, assuming certain generic conditions. We will
also discuss some applications of this result in general relativity.

####
**
May 17, 2022 Tue, 3:00-4:00PM
**

Natasa Sesum, Rutgers

**TBD:
**
TBD.

####
**
May 24, 2022 Tue, 3:00-4:00PM
**

Alex Mramor, John Hopkins

**TBD:
**
TBD.

####
**
May 31, 2022 Tue, 3:00-4:00PM
**

Misha Karpukhin, California Institute of Technology

**TBD:
**
TBD.

### Winter 2022

####
**
January 18, 2022 Tue, 3:00-4:00PM
**

Zhichao Wang, UBC

**Uryson width of three dimensional mean convex domains with non-negative Ricci curvature:
**
In this joint work with B. Zhu, we prove that for every three dimensional manifold with nonnegative Ricci
curvature and strictly mean convex boundary, there exists a Morse function so that each connected component of its
level sets has a uniform diameter bound, which depends only on the lower bound of mean curvature. This gives an
upper bound of Uryson 1-width for those three manifolds with boundary.

####
**
January 25, 2022 Tue, 3:00-4:00PM
**

Mariel Saez, Pontificia Universidad Catolica de Chile

**Uniqueness of entire graphs evolving by mean curvature flow:
**
In this talk I will discuss the uniqueness of graphical mean curvature flow. We consider as initial
conditions graphs of locally Lipschitz functions and prove that in the one dimensional case solutions are unique
without any further assumptions. This result is then generalized for rotationally symmetric solutions. In the
general n- dimensional case, we prove uniqueness under additional conditions: we require a uniform lower bound on
the second fundamental form and the height function of the initial condition. The latter result extends to initial
conditions that are proper graphs over subdomains of Rn. (Joint with P. Daskalopoulos).

####
**
February 1, 2022 Tue, 3:00-4:00PM
**

Jiayin Pan, Fields Institute

**The escape phenomenon in open manifolds with nonnegative Ricci curvature:
**
We start with a simple phenomenon in an open manifold M with nonnegative Ricci curvature: the set of all minimal
representing geodesic loops of pi_1(M,p) may not be contained in any bounded sets of M. We will talk about how
this escape phenomenon is related to the group structure of fundamental groups and the metric structure of
asymptotic cones. Part of this talk is joint work with Guofang Wei.

####
**
February 8, 2022 Tue, 3:00-4:00PM
**

Xin Zhou, Cornell

**Min-max minimal hypersurfaces with multiplicity two:
**
In this talk, we will exhibit a set of non-bumpy metrics on the standard (n+1)-sphere, in which the min-max
minimal hypersurface associated with the second volume spectrum is a multiplicity two n-sphere. Such non bumpy
metrics form the first set of examples where the min-max theory must produce higher multiplicity minimal
hypersurfaces. This is a joint work with Zhichao Wang.

####
**
February 15, 2022 Tue, 3:00-4:00PM
**

Nick Edelen, Notre Dame

**Degeneration of 7-dimensional minimal hypersurfaces with bounded index:
**
A 7D minimal and locally-stable hypersurface will in general have a discrete singular set, provided it has no
singularities modeled on a union of half-planes. We show in this talk that the geometry/topology/singular set of
these surfaces has uniform control, in the following sense: if $M_i$ is a sequence of 7D minimal hypersurfaces
with uniformly bounded index and area, and discrete singular set, then up to a subsequence all the $M_i$ are
bi-Lipschitz equivalent, with uniform Lipschitz bounds on the maps. As a consequence, we prove the space of $C^2$
embedded minimal hypersurfaces in a fixed $8$-manifold, having index $\leq I$, area $\leq \Lambda$, and discrete
singular set, divides into finitely-many diffeomorphism types.

####
**
February 22, 2022 Tue, 3:00-4:00PM
**

Pei-Ken Hung, University of Minnesota

**Toroidal positive mass theorem:
**
I will discuss the positive mass theorem for 3-dimensional asymptotically hyperboloidal initial data sets with
toroidal infinity. The rigidity is obtained provided the second fundamental form is umbilic. Without the umbilic
assumption, the rigidity fails and we construct a counterexample based on the Horowitz-Myers geon. We also recover
the local rigidity theorem of Eichmair-Galloway-Mendes in dimension 3. These results are obtained through an
analysis of spacetime harmonic functions. This is a joint work with Aghil Alaee and Marcus Khuri.

####
**
March 1, 2022 Tue, 3:00-4:00PM
**

Ovidiu Munteanu, University of Connecticut

**Comparison results for complete noncompact three-dimensional manifolds:
**
Typical comparison results in Riemannian geometry, such as for volume or for spectrum of the Laplacian, require
Ricci curvature lower bounds. In dimension three, we can prove some sharp comparison estimates assuming only a
scalar curvature bound. The talk will present these results, their applications, and explain how dimension three
is used in the proofs.

####
**
March 8, 2022 Tue, 3:00-4:00PM
**

Alex Waldron, the University of Wisconsin

**Strict type-II blowup in harmonic map flow:
**
I'll describe some recent work on 2D harmonic map flow, in which I show that a familiar bound on the blowup rate
at a finite-time singularity is sufficient for continuity of the body map. This is relevant to a conjecture of
Topping.

####
**
March 15, 2022 Tue, 3:00-4:00PM
**

Lucas Ambrozio, IMPA

**Analogues of Zoll metrics in minimal submanifold theory:
**
A Riemannian metric on a closed manifold is called Zoll when all of its geodesics are closed and have the same
period. Non-trivial rotationally symmetric Zoll metrics on the two-sphere were constructed by Otto Zoll in the
beginning of the 1900's, but many questions about these metrics are still open. It seems that higher-dimensional
analogues of Zoll metrics, where closed geodesics are replaced by closed embedded minimal hypersurfaces, could be
very interesting objects to be investigated in relation to isodiastolic inequalities and other geometric problems,
but also on their own account. In this talk, I will discuss some recent results about the construction and
geometric understanding of these new Zoll-like geometries. This is a joint project with F. Marques (Princeton) and
A. Neves (UChicago).

### Autumn 2021

####
**
September 28, 2021 Tue, 5:30-6:30PM
Note the special time this week!
**

Otis Chodosh, Stanford

**Stable minimal hypersurfaces in $\mathbb{R}^4$:
**
I will explain why a complete two-sided stable minimal hypersurface in $\mathbb{R}^4$ is flat. This is joint work
with Chao Li.

####
**
October 5, 2021 Tue, 3:00-4:00PM
**

Antoine Song, UC Berkeley

**
A quantification of Gromov's nonsqueezing theorem in dimension 4:
**
The nonsqueezing theorem states that a ball B_R of radius larger than 1 in R^4 (with the standard symplectic
structure) cannot be symplectically embedded inside the cylinder D x R^2 where D is the unit 2-disk. I will
explain that this result can be quantified as follows: if E is a closed set and the complement of E in B_R
symplectically embeds inside D x R^2, then the Minkowski dimension of E is at least 2, and this is optimal in
general. The proof uses Gromov's waist inequality. This is joint work with Kevin Sackel, Umut Varolgunes and
Jonathan Zhu.

####
**
October 12, 2021 Tue, 3:00-4:00PM
**

Lu Wang, Yale

**
Closed hypersurfaces of low entropy are isotopically trivial:
**
We show that any closed connected hypersurface in four-dimensional Euclidean space with entropy less than or equal
to that of the round cylinder is smoothly isotopic to the standard three-sphere. This is joint work with Jacob
Bernstein.

####
**
October 19, 2021 Tue, 3:00-4:00PM
**

Martin Lesourd, Harvard

**R>0 on noncompact manifolds - topology, geometry, and mass:
**
"Which manifolds admit a complete metric of positive scalar curvature R>0?" has gathered great interest over the
years. The question has been solved for certain kinds of closed manifolds, but the story for noncompact manifolds
seems to be more subtle. A related question (emphasized by Gromov among others) is "What is the geometry of spaces
with R>0?" Both of these questions are related to the positive mass theorem for asymptotically flat manifolds. The
talk will describe this connection along with some recent work with Ryan Unger and S-T Yau on the positive mass
theorem with arbitrary ends.

####
**
October 26, 2021 Tue, 3:00-4:00PM
**

Keaton Naff, MIT

**A local noncollapsing estimate for mean curvature flow:
**
We will discuss noncollapsing in mean curvature flow and then discuss a localization of the noncollapsing
estimate. By combining our result with earlier work of X.-J. Wang, it follows that certain ancient convex
solutions that sweep out the entire space are noncollapsed. This is joint work with Simon Brendle.

####
**
November 2, 2021 Tue, 3:00-4:00PM
**

Theodora Bourni, University of Tennessee Knoxville

**Collapsed and non-collapsed ancient solutions to mean curvature flow:
**
Mean curvature flow (MCF) is the gradient flow of the area functional; it moves the surface in the direction of
steepest decrease of area. An important motivation for the study of MCF comes from its potential geometric
applications, such as classification theorems and geometric inequalities. MCF develops "singularities" (curvature
blow-up), which obstruct the flow from existing for all times and therefore understanding these high curvature
regions is of great interest. This is done by studying ancient solutions, solutions that have existed for all
times in the past, and which model singularities. In this talk we will present recent developments concerning
ancient solutions. Among other things we will discuss a new way of constructing and classifying collapsed
solutions. This is joint work with Mat Langford, Stephen Lynch and Giuseppe Tinaglia.

####
**
Novbember 9, 2021 Tue, 3:00-4:00PM
**

Franco Vargas-Pallete, Yale

**Stable minimal hypersurfaces in $\mathbb{R}^4$:
**
Isoperimetric profiles for convex-cocompact hyperbolic 3-manifolds : An isoperimetric profile of a Riemannian
manifold is a function that for each positive number $V$ associates the optimal perimeter needed bound a volume
equal to $V$. On this talk we'll see how for convex co-compact hyperbolic 3-manifolds this relates to Renormalized
Volume (a studied functional on the deformation space). We will use this relation together with some tools from
General relativity (namely the Hawking mass) to prove that, in the appropriate setup, the isoperimetric profile of
a hyperbolic 3-manifold stays below the profile of a model, and equality occurs if and only if the manifold is
isometric to the model. This is joint work with Celso Viana.

####
**
Novbember 16, 2021 Tue, 3:00-4:00PM
**

Akashdeep Dey, Princeton

**A comparison of the Almgren-Pitts and the Allen-Cahn min-max theory:
**
Min-max theory for the Allen-Cahn equation was developed by Guaraco and Gaspar-Guaraco. Allen-Cahn min-max theory
provides a PDE based alternative to the Almgren-Pitts min-max theory for the min-max construction of minimal
hypersurfaces. In my talk, I will briefly describe the Almgren-Pitts and the Allen-Cahn min-max theory and discuss
the question to what extent these two theories agree. Part of the talk will be based on the above mentioned works
of Guaraco and Gaspar-Guaraco.

####
**
Novbember 30, 2021 Tue, 3:00-4:00PM
**

Ben Lowe, Princeton

**Area, Scalar Curvature, and Hyperbolizable Riemannian 3-Manifolds:
**
On any closed hyperbolizable 3-manifold, we find a sharp relation between the minimal surface entropy (introduced
by Calegari-Marques-Neves) and the average area ratio (introduced by Gromov), and we show that, among metrics g
with scalar curvature greater than or equal to -6, the former is maximized by the hyperbolic metric. One corollary
is to solve a conjecture of Gromov regarding the average area ratio. We also prove a sharp lower bound for the
area in $g$ of a surface homotopic to a totally geodesic surface in the hyperbolic metric. Our proofs use Ricci
flow with surgery and laminar measures invariant under a PSL(2,R)-action. Most of the talk will be based on the
paper (joint with Andre Neves) https://arxiv.org/abs/2110.09451.

####
**
December 7, 2021 Tue, 3:00-4:00PM
**

Nick Edelen, Notre Dame

**TBA:
**
TBA

### Spring 2021

####
**
April 13, 2021 Tue, 1:00-2:00PM
**

Salvatore Stuvard, UT Austin

**The regularity of mass minimizing currents modulo p:
**
Integer rectifiable currents mod(p) are a class of generalized surfaces in which it is possible to define and
solve Plateau's problem. The corresponding minimizers, mass minimizing currents mod(p), exhibit a far richer
geometric complexity than the classical mass minimizing integral currents of Federer and Fleming. In this talk, I
will present the partial interior regularity theory for these objects. The focus will be on dimension bounds and
fine structural properties (such as rectifiability and local finiteness of measure) of their singular sets. The
ultimate goal will be to reveal that singularities of mass minimizing currents mod(p) present an interesting
regular free boundary structure. This is based on multiple joint works with Camillo De Lellis (IAS), Jonas Hirsch
(U Leipzig), Andrea Marchese (U Trento), and Luca Spolaor (UCSD).

####
**
April 20, 2021 Tue, 1:00-2:00PM
**

Gabor Szekelyhidi, Notre Dame

**Uniqueness of certain cylindrical tangent cones:
**
Leon Simon showed that if an area minimizing hypersurface admits a cylindrical tangent cone of the form C x R,
then this tangent cone is unique for a large class of minimal cones C. One of the hypotheses in this result is
that C x R is integrable and this excludes the case when C is the Simons cone over S^3 x S^3. The main result in
this talk is that the uniqueness of the tangent cone holds in this case too. The new difficulty in this
non-integrable situation is to develop a version of the Lojasiewicz-Simon inequality that can be used in the
setting of tangent cones with non-isolated singularities.

####
**
April 27, 2021 Tue, 1:00-2:00PM
**

Yi Lai, UC Berkeley

**A family of 3d steady gradient Ricci solitons that are flying wings:
**
We find a family of 3d steady gradient Ricci solitons that are flying wings. This verifies a conjecture by
Hamilton. For a 3d flying wing, we show that the scalar curvature does not vanish at infinity. The 3d flying wings
are collapsed. For dimension n Ã¢ÂÂ¥ 4, we find a family of Z2 ÃÂ O(n Ã¢ÂÂ 1)-symmetric but non-rotationally
symmetric n-dimensional steady gradient solitons with positive curvature operator. We show that these solitons are
non-collapsed.

####
**
May 4, 2021 Tue, 1:00-2:00PM
**

Yangyang Li, Princeton

**Generic Regularity of Minimal Hypersurfaces in Dimension 8:
**
The well-known Simons' cone suggests that minimal hypersurfaces could be possibly singular in a Riemannian
manifold with dimension greater than 7, unlike the low dimensional case. Nevertheless, it was conjectured that one
could perturb away these singularities generically. In this talk, I will discuss how to perturb them away to
obtain a smooth minimal hypersurface in an 8-dimension closed manifold, by induction on the "capacity" of singular
sets. This result generalizes the previous works by N. Smale and by Chodosh-Liokumovich-Spolaor to any
8-dimensional closed manifold. This talk is based on joint work with Zhihan Wang.

####
**
May 11, 2021 Tue, 1:00-2:00PM
**

Davi Maximo, University of Pennsylvania

**The Waist Inequality and Positive Scalar Curvature:
**
The topology of three-manifolds with positive scalar curvature has been (mostly) known since the solution of the
Poincare conjecture by Perelman. Indeed, they consist of connected sums of spherical space forms and S^2 x S^1's.
In spite of this, their "shape" remains unknown and mysterious. Since a lower bound of scalar curvature can be
preserved by a codimension two surgery, one may wonder about a description of the shape of such manifolds based on
a codimension two data (in this case, 1-dimensional manifolds). In this talk, I will show results from a recent
collaboration with Y. Liokumovich elucidating this question for closed three-manifolds.

####
**
May 25, 2021 Tue, 1:00-2:00PM
**

Renato Bettiol, CUNY

**Minimal 2-spheres in ellipsoids of revolution:
**
Motivated by Morse-theoretic considerations, Yau asked in 1987 whether all minimal 2-spheres in a 3-dimensional
ellipsoid inside R^4 are planar, i.e., determined by the intersection with a hyperplane. Recently, this was shown
not to be the case by Haslhofer and Ketover, who produced an embedded non-planar minimal 2-sphere in sufficiently
elongated ellipsoids, combining Mean Curvature Flow and Min-Max methods. Using Bifurcation Theory and the
symmetries that arise if at least two semi-axes coincide, we show the existence of arbitrarily many distinct
embedded non-planar minimal 2-spheres in sufficiently elongated ellipsoids of revolution. This is based on joint
work with P. Piccione.

####
**
June 1, 2021 Tue, 10:00-11:00AM
**

Jaigyoung Choe, Korea Institute for Advanced Study

**The periodic Plateau problem and its application:
**
The periodic Plateau problem will be proposed and solved. As an application it will be proved that there exist
four minimal annuli in a tetrahedron which are perpendicular to its faces. Also it will be proved that every
Platonic solid contains three minimal surfaces of genus 0 perpendicular to its faces.

### Winter 2021

####
**
January 19, 2021 Tue, 1:00-2:00PM
**

Hans-Joachim Hein, MÃÂ¼nster

**Smooth asymptotics for collapsing Calabi-Yau metrics:
**
Yau's solution of the Calabi conjecture provided the first nontrivial examples of Ricci-flat Riemannian
metrics on compact manifolds. Attempts to understand the degeneration patterns of these metrics in families have
revealed many remarkable phenomena over the years. I will review some aspects of this story and present recent
joint work with Valentino Tosatti where we obtain a complete asymptotic expansion (locally uniformly away from the
singular fibers) of Calabi-Yau metrics collapsing along a holomorphic fibration of a fixed compact Calabi-Yau
manifold. This relies on a new analytic method where each additional term of the expansion arises as the
obstruction to proving a uniform bound on one additional derivative of the remainder.

####
**
January 26, 2021 Tue, 1:00-2:00PM
**

Fritz Hiesmayr, UCL

**An Urbano-type theorem for the Allen-Cahn equation:
**
The Allen-Cahn equation is a semilinear elliptic PDE modelling phase transitions in two-phase media. In recent
years this has found applications in geometry due to its link with minimal hypersurfaces. We present an analogue
of Urbano's theorem about minimal surfaces in the round three-sphere. Our result is a rigidity theorem for
solutions of the Allen-Cahn equation in S^3 with small index. They are symmetric, and vanish either on a minimal
sphere or a Clifford torus. One key observation is that the nodal sets of two distinct solutions must have
non-empty intersection.

####
**
February 2, 2021 Tue, 1:00-2:00PM
**

Daren Cheng, Waterloo

**Existence of constant mean curvature 2-spheres in Riemannian 3-spheres:
**
In this talk I'll describe recent joint work with Xin Zhou, where we make progress on the question of finding
closed constant mean curvature surfaces with controlled topology in 3-manifolds. We show that in a 3-sphere
equipped with an arbitrary Riemannian metric, there exists a branched immersed 2-sphere with constant mean
curvature H for almost every H. Moreover, the existence extends to all H when the target metric is positively
curved. This latter result confirms, for the branched immersed case, a conjecture of Harold Rosenberg and Graham
Smith.

####
**
February 9, 2021 Tue, 1:00-2:00PM
**

Aleksander Doan, Columbia

**Counting pseudo-holomorphic curves in symplectic six-manifolds:
**
The number of embedded pseudo-holomorphic curves in a symplectic manifold typically depends on the choice of
an almost complex structure on the manifold and so does not lead to a symplectic invariant. However, I will
discuss two instances in which such naive counting does define a symplectic invariant. The proof of invariance
combines methods of symplectic geometry with results of geometric measure theory, especially regularity theory for
calibrated currents. The talk is based on joint work with Thomas Walpuski. Time permitting, I will also discuss a
related project, joint with Eleny Ionel and Thomas Walpuski, whose goal is to use geometric measure theory to
prove the Gopakumar-Vafa finiteness conjecture.

####
**
February 16, 2021 Tue, 1:00-2:00PM
**

Costante Bellettini, UCL

**Allen-Cahn minmax and multiplicity-1 minimal hypersurfaces**
The existence of a closed minimal hypersurface in a compact Riemannian manifold was first established by the
combined efforts of Almgren, Pitts, Schoen-Simon-Yau, Schoen-Simon in the early 80s by means of what is nowadays
called minmax a la Almgren-Pitts. An alternative approach to reach the same existence result has been implemented
in recent years in a work by Guaraco, using a minmax construction for the Allen-Cahn energy, in combination with
works by Hutchinson-Tonegawa, Tonegawa, Tonegawa-Wickramasekera, Wickramasekera. A natural question (ubiquitous in
geometric analysis and, in particular, in minmax constructions) is whether the minimal hypersurface is obtained
with multiplicity 1. The multiplicity-1 information has important geometric consequences. However, the a priori
possibility of higher multiplicity is intrinsic in both minmax constructions, as they are carried out in the class
of varifolds. After an overview, this talk focuses on the case of an ambient Riemannian manifold (of dimension 3
or higher) with positive Ricci curvature: in this case, the minmax construction via Allen-Cahn yields a
multiplicity-1 minimal hypersurface. If time permits, the case of low-dimensional manifolds endowed with a bumpy
metric will also be addressed.

####
**
February 23, 2021 Tue, 1:00-2:00PM
**

Robin Neumayer, Northwestern

**$d_p$ Convergence and $\epsilon$-regularity theorems for entropy and scalar curvature lower bounds:**
In this talk, we consider Riemannian manifolds with almost non-negative scalar curvature and Perelman entropy.
We establish an $\epsilon$-regularity theorem showing that such a space must be close to Euclidean space in a
suitable sense. Interestingly, such a result is false with respect to the Gromov-Hausdorff and Intrinsic Flat
distances, and more generally the metric space structure is not controlled under entropy and scalar lower bounds.
Instead, we introduce the notion of the $d_p$ distance between (in particular) Riemannian manifolds, which
measures the distance between $W^{1,p}$ Sobolev spaces, and it is with respect to this distance that the
$\epsilon$ regularity theorem holds. We will discuss various applications to manifolds with scalar curvature and
entropy lower bounds, including a compactness and limit structure theorem for sequences and a priori $L^p$ scalar
curvature bounds for $p < 1$. This is joint work with Man-Chun Lee and Aaron Naber.

####
**
March 2, 2021 Tue, 1:00-2:00PM
**

Yevgeny Liokumovich, Toronto

**Generic regularity of min-max minimal hypersurfaces:
**
Minimal hypersurfaces in 8-dimensional Riemannian manifolds may have isolated singularities. However, it follows
from results of R. Hardt, L. Simon and N. Smale that one can perturb away singularities of an area minimizing
minimal hypersurface by a small change of the metric. I will talk about a similar problem for min-max minimal
hypersurfaces (joint work with Otis Chodosh and Luca Spolaor). We show that for a generic 8-dimensional
Riemannian manifold with positive Ricci curvature, there exists a smooth minimal hypersurface. Without the
curvature condition, we show that for a dense set of 8-dimensional Riemannian metrics there exists a minimal
hypersurface with at most one singular point. Our proof uses a construction of optimal nested sweepouts from a
joint work with Gregory Chambers.

####
**
March 9, 2021 Tue, 1:00-2:00PM
**

Tristan Ozuch, MIT

**Higher order obstructions to the desingularization of Einstein metrics:
**
We exhibit new obstructions to the desingularization of Einstein metrics in dimension 4. These obstructions are
specific to the compact situation and raise the question of whether or not a sequence of compact Einstein
metrics degenerating while bubbling out gravitational instantons has to be KÃÂ¤hler-Einstein. We then test
these obstructions to discuss the possibility of producing a Ricci-flat but not Kahler metric by the promising
desingularization configuration proposed by Page in 1981.

### Autumn 2020

####
**
October 6, 2020 Tue, 11:00am-12pm
**

Dmitry Jakobson, McGill

**
Zero and negative eigenvalues of conformally covariant operators, and nodal sets in conformal geometry:
**
We first describe conformal invariants that arise from nodal sets and negative eigenvalues of conformally
covariant operators (such as Yamabe or Paneitz operator). We discuss applications to curvature prescription
problems. We prove that the Yamabe operator can have an arbitrarily large number of negative eigenvalues on any
manifold of dimension greater than 2. We show that 0 is generically not an eigenvalue of the conformal
Laplacian. If time permits, we shall discuss related results on manifolds with boundary, as well as for weighted
graphs. This is joint work with Y. Canzani, R. Gover, R. Ponge, A. Hassannezhad, M. Levitin, M. Karpukhin, G.
Cox and Y. Sire.

####
**
October 13, 2020 Tue, 11:00am-12pm**

Christos Mantoulidis, Brown

**
Ancient mean curvature flows, gradient flows, and Morse index:
**
This talk gives an overview of two recent joint works. The first work, joint with Kyeongsu Choi, studies closed
ancient solutions to gradient flows of elliptic functionals in Riemannian manifolds. Our methods classify
ancient solutions coming out of a critical point, with very mild decay assumptions, establishing that they are
parametrized by unstable eigenfunctions of the critical point. Applied to mean curvature flow, these methods
imply an arbitrary dimension and codimension classification of ancient mean curvature flows of closed
submanifolds of S^n with low area. The second work, joint with Otis Chodosh, Kyeongsu Choi, and Felix Schulze,
extends these methods to study ancient mean curvature flows lying on one side of asymptotically conical
shrinking solitons. As an application, we partially confirm the conjectured non-linear instability of
asymptotically conical singularity models in mean curvature flow.

####
**
October 20, 2020 Tue, 5:00pm-6:00pm
**

Jonathan Zhu, Princeton

**
Explicit Åojasiewicz inequalities for mean curvature flow shrinkers:
**
Åojasiewicz inequalities are a popular tool for studying the stability of geometric structures. For mean
curvature flow, Schulze used Simon's reduction to the classical Åojasiewicz inequality to study compact tangent
flows. Colding and Minicozzi instead used a direct method to prove Åojasiewicz inequalities for round
cylinders. We'll discuss similarly explicit Åojasiewicz inequalities and applications for Clifford shrinkers
and certain other cylinders.

####
**
October 27, 2020 Tue, 11:00am-12:00pm**

Robert Haslhofer, University of Toronto

**
Ancient flows and the mean-convex neighborhood conjecture:
**
I will explain our recent proof of the mean-convex neighborhood conjecture. The key is a classification result
for ancient asymptotically cylindrical mean curvature flows. The 2-dimensional case is joint work with Choi and
Hershkovits, and the higher-dimensional case is joint with Choi, Hershkovits and White.

####
**
November 3, 2020 Tue, 11:00am-12:00pm
**

GonÃ§alo Oliveira, Universidade Federal Fluminense

**
G2-monopoles (a summary):
**
This talk is aimed at reviewing what is known about G2-monopoles and motivate their study. After this, I will
mention some recent results obtained in collaboration with Ãkos Nagy and Daniel Fadel which investigate the
asymptotic behavior of G2-monopoles. Time permitting, I will mention a few possible future directions regarding
the use of monopoles in G2-geometry.

####
**
November 10, 2020 Tue, 11:00am-12:00pm
**

Ao Sun, University of Chicago

**
Generic Mean Curvature Flow and Generalizations of Entropy:
**
Mean curvature flow entropy was introduced by Colding-Minicozzi, and it is a very important quantity in the
study of mean curvature flow and related geometric problem. In particular, mean curvature flow entropy plays a
crucial role in the study of generic mean curvature flow. I will discuss two generalizations of mean curvature
flow entropy: one is a localized version of entropy, another one is entropy in a closed manifold. I will discuss
how to use these generalizations of entropy to rule out some pathological asymptotic behaviors of mean curvature
flow.

####
**
November 17, 2020 Tue, 11:00am-12:00pm
**

Liam Mazurowski, University of Chicago

**
CMC doublings of minimal surfaces via min-max:
**
An interesting problem in differential geometry is to try to understand the space of constant mean curvature
surfaces (CMCs) in a given manifold. Recently Zhou and Zhu developed a min-max theory for constructing CMCs, and
used it to show that any manifold M of dimension between 3 and 7 contains a smooth, almost-embedded CMC
hypersurface of mean curvature h for every h>0. In this talk, I will explain how this min-max theory can be used
to construct CMC doublings of certain minimal surfaces in 3-manifolds. Such CMC doublings were previously
constructed for minimal hypersurfaces in M^n with n>3 by Pacard and Sun using gluing methods.

####
**
November 24, 2020 Tue, 11:00am-12:00pm
**

Shubham Dwivedi, Humboldt University of Berlin

**
Deformation theory of nearly G_2 manifolds:
**
We will discuss the deformation theory of nearly G_2 manifolds. After defining nearly G_2 manifolds, we will
describe some identities for 2 and 3 forms on such manifolds. We will introduce a Dirac type operator which will
be used to prove new results on the cohomology of nearly G_2 manifolds. Along the way we will reprove a result
of Alexandrov-Semmelman on the space of infinitesimal deformation of nearly G_2 structures. Finally, we will
prove that the infinitesimal deformations of the homogeneous nearly G_2 structure on the Aloff--Wallach space
are obstructed to second order. The talk is based on a joint work with Ragini Singhal (University of Waterloo).

####
**
December 1, 2020 Tue, 11:00am-12:00pm
**

Alexandre Girouard, Universite de Laval

**
Optimal isoperimetric upper bounds for Steklov eigenvalues of planar domains:
**
Given a compact Riemannian manifold M, the eigenvalues of the Dirichlet-to-Neumann map acting on the boundary of
M are known as Steklov eigenvalues of M. I will present a complete solution of the isoperimetric problem for
each perimeter-normalized Steklov eigenvalue of planar domains: the best upper bound for the $k+1$-th
perimeter-normalized Steklov eigenvalue is $8k\pi$. The proofs are based on continuity properties for
variational eigenvalues associated to Radon measures on compact manifolds. In particular, starting with any
domain in the plane, we constructed a sequence of subdomains with its $k+1$-th perimeter-normalized Steklov
eigenvalue converging to $8k\pi$. These subdomains are obtained by removing small disks from the initial domain,
in the spirit of homogenization theory. Properties of maximizing sequences will also be discussed, showing in
particular that any sequence of domains with prescribed perimeter with its first nonzero Steklov eigenvalue
converging to $8\pi$ must collapse to a point. This talk is based on recent papers with with Antoine Henrot (U.
de Lorraine), Mikhail Karpukhin (Caltech) and Jean LagacÃ© (UCL).

### Winter 2020

####
**
January 14, 2020 Tue, 3:30-4:30PM, Eckhart 202**

Dan Lee, CUNY

**
Progress on Bartnik's stationary conjecture:
**
Given a compact initial data set with boundary, the Bartnik problem is the problem of finding asymptotically
flat initial data that extends given data in such a way that it minimizes mass among all possible (admissible)
extensions satisfying the dominant energy condition. Bartnik conjectured that if such a minimizer exists, then
it must be stationary in the extended region. In the time-symmetric case, this was settled by Corvino (in which
case the stationary extension is actually static). In joint work with Lan-Hsuan Huang of the University of
Connecticut, we are able to prove in the general case that a minimizing extension must be vacuum stationary
outside some large compact set. Our proof involves finding ways to locally deform the geometry of an initial
data set in such a way that we have useful control over the energy-momentum density.

####
**
January 21, 2020 Tue, 3:30-4:30PM, Eckhart 202
**

Gigliola Staffilani, MIT

**
Some results on the almost everywhere convergence of the Schrodinger flow:
**
In this work we are concerned with the question of almost everywhere convergence of the nonlinear Schrodinger
flow as time tends to zero, both in the continuous and the periodic case. We will review the extraordinary
progress made in the linear continuous case and we will illustrate some progress recently made in the nonlinear
case using both a deterministic and a probabilistic approach. This is joint work with E. Compaan and R. Luca.

####
**
January 21, 2020 Tue, 4:30-5:30PM, Ryerson 276
**

Mohammad Ghomi, Georgia Tech

**
Isoperimetric inequality in spaces of nonpositive curvature:
**
The classical isoperimetric inequality states that in Euclidean space spheres provide unique enclosures of least
perimeter for any given volume. In this talk we show that this inequality generalizes to spaces of nonpositive
curvature, or Cartan-Hadamard manifolds, as conjectured by Aubin, Gromov, Burago, and Zalgaller in 1970s and
80s. The proof is based on a comparison formula for total curvature of level sets in Riemannian manifolds. This
is joint work with Joel Spruck.

####
**
January 31, 2020 Fri, 2:00-3:00PM, Eckhart 207
**

Philippe G. LeFloch, Sorbonne University and CNRS

**
Nonlinear stability of self-gravitating matter under low decay and weak regularity conditions:
**
I will present recent progress on the global evolution problem for self-gravitating matter. (1) For Einstein's
constraint equations, motivated by a scheme proposed by Carlotto and Schoen I will show the existence of
asymptotically Euclidean Einstein spaces with low decay; joint work with T. Nguyen. (2) For Einstein's evolution
equations in the regime near Minkowski spacetime, I will show the global nonlinear stability of massive matter
fields; joint work with Y. Ma. (3) For the colliding gravitational wave problem, I will show the existence of
weakly regular spacetimes containing geometric singularities across which junction conditions are imposed; joint
work with B. Le Floch and G. Veneziano.

####
**
February 4, 2020 Tue, 3:30-4:30PM, Eckhart 202
**

Pedro Gaspar, UChicago

**
Solutions of the Allen-Cahn equation on closed manifolds in the presence of symmetry:
**
In recent years, there has been remarkable development about the connection between the Allen-Cahn equation and
minimal hypersurfaces. A result due to F. Pacard and M. RitorÃ shows that given a minimal hypersurface Î in a
closed manifold M, we can find solutions of this PDE whose nodal sets approximate Î, provided it is a
nondegenerate critical point of the area functional. This condition is often too restrictive, as many
interesting examples of ambient manifolds have continuous groups of symmetries, and the corresponding variations
preserve the area of any hypersurface. In this talk, we explain how to obtain a similar existence result in the
case where all Jacobi fields of Î arise from ambient isometries. We also derive some geometric consequences,
and describe a second order convergence result for multiplicity-one solutions, in terms of the corresponding
linearized operator and its eigenvalues. This is joint work with R. Caju.

####
**
February 11, 2020 Tue, 3:30-4:30PM, Eckhart 202
**

Celso dos Santos Viana, UC Irvine

**
Isoperimetry and volume preserving stability in real projective spaces:
**
In this talk I will address the problem of classifying volume preserving stable constant mean curvature
hypersurfaces in Riemannian manifolds. I will present recent classification in the real projective space of any
dimension and, consequently, the solution of the isoperimetric problem.

####
**
February 18, 2020 Tue, 3:30-4:30PM, Eckhart 202
**

Antoine Song, UC Berkeley

**
Morse index, Betti numbers and singular set of bounded area minimal hypersurfaces:
**
We will explain how, for minimal hypersurfaces with uniformly bounded area, the topology and the singular set
can be controlled by the Morse index. We use this to study the sequence of minimal surfaces constructed by
min-max theory in a given closed 3-manifold. Some natural open questions will be introduced.

####
**
February 25, 2020 Tue, 3:30-4:30PM, Eckhart 202
**

Peter McGrath, UPenn

**
Generalizing the Linearized Doubling Approach and New Minimal Surfaces and Self-Shrinkers via Doubling:
**
I will discuss recent work (with N. Kapouleas) on generalizing the Linearized Doubling approach to apply (under
reasonable assumptions) to doubling arbitrary closed minimal surfaces in arbitrary Riemannian three-manifolds
without any symmetry requirements. More precisely, given a family of LD solutions on a closed minimal surfaces
embedded in a Riemannian three-manifold, where an LD solution is a solution of the Jacobi equation with
logarithmic singularities, we prove a general theorem which states that if the family satisfies certain
conditions, then a new minimal surface can be constructed via doubling, with catenoidal bridges replacing the
singularities of one of the LD solutions. The construction of the required LD solutions is currently only
understood when the surface and ambient manifold possess O(2) symmetry and the number of bridges is chosen large
enough along O(2) orbits. In this spirit, we use the theorem to construct new self-shrinkers of the mean
curvature flow via doubling the spherical self-shrinker and new complete embedded minimal surfaces of finite
total curvature in the Euclidean three-space via doubling the catenoid.

####
**
March 10, 2020 Tue, 3:30-4:30PM, Eckhart 202
**

Ruobing Zhang, Stony Brook

**
TBA:
**
TBA

### Autumn 2019

####
**
October 8, 2019 Tue, 3:30-4:30PM, Eckhart 202
**

Antonio De Rosa, Courant Institute, NYU

**
Elliptic integrands in geometric analysis:
**
We present the recent tools we developed to prove existence and regularity properties of the critical points of
anisotropic functionals. In particular, we present our extension of Allard's celebrated rectifiability theorem
to the setting of varifolds with bounded anisotropic first variation. We apply this result to solve the
set-theoretic anisotropic Plateau problem. We obtain as corollaries easy solutions to three different
formulations of the Plateau problem, introduced by Reifenberg, by Harrison-Pugh and by David. Moreover, we prove
an anisotropic counterpart of Allard's compactness theorem for integral varifolds. To conclude, we focus on the
anisotropic isoperimetric problem. We prove the anisotropic counterpart of Alexandrov's characterization of
volume-constrained critical points among finite perimeter sets. Furthermore we derive stability inequalities
associated to this rigidity theorem. Some of the presented theorems are joint works with De Lellis, De
Philippis, Ghiraldin, Gioffre, Kolasinski and Santilli.

####
**
October 15, 2019 Tue, 3:30-4:30PM, Eckhart 202
**

Daren Cheng, UChicago

**
Bubble tree convergence of conformally cross product preserving maps:
**
We study a class of weakly conformal maps, called Smith maps, which parametrize associative 3-folds in
7-manifolds equipped with G2-structures. These maps satisfy a system of first-order PDEs generalizing the
Cauchy-Riemann equation for J-holomorphic curves, and we are interested in their bubbling phenomena.
Specifically, we first prove an epsilon-regularity theorem for Smith maps in W^{1, 3}, and then explain how that
combines with conformal invariance to yield bubble trees of Smith maps from sequences of such maps with
uniformly bounded 3-energy. When the G2-structure is closed, we show that both the 3-energy and the homotopy are
preserved in the bubble tree limit. The result can be viewed as an associative analogue of the bubble tree
convergence theorem for J-holomorphic curves. This is joint work with Spiro Karigiannis and Jesse Madnick.

####
**
October 22, 2019 Tue, 3:30-4:30PM, Eckhart 202
**

Zheng Huang, CUNY-Staten Island

**
Bifurcation for closed minimal surfaces in hyperbolic three-manifolds:
**
In the 1970s, Uhlenbeck initiated a program to study questions about existence, multiplicity of closed minimal
surfaces in hyperbolic three-manifolds as well as applications in various related areas. In this talk, based on
joint work with M. Lucia and G. Tarantello, I will describe several results in these directions.

####
**
October 29, 2019 Tue, 3:30-4:30PM, Eckhart 202
**

Xiaodong Wang, Michigan State

**
On the size of compact Riemannian manifolds with nonnegative Ricci curvature and convex boundary:
**
Given a compact Riemannian manifold with nonnegative Ricci curvature and convex boundary it is interesting to
estimate its size in terms of the volume, the area of its boundary etc. I will discuss some open problems and
present some partial results.

####
**
November 5, 2019 Tue, 3:30-4:30PM, Eckhart 202
**

Mikhail Karpukhin, UC Irvine

**
Isoperimetric inequalities for Laplacian eigenvalues: recent developments:
**
The Laplacian is a canonical second order elliptic operator defined on any Riemannian manifold. The study of
upper bounds for its eigenvalues under the volume constraint is a classical problem of spectral geometry going
back to J. Hersch, P. Li and S.-T. Yau. The particular interest to this problem stems from a surprising
connection to the theory of minimal surfaces in spheres. In the present talk we will survey some recent results
in the area with an emphasis on the role played by the index of minimal surfaces. In particular, we will discuss
some recent applications, including a new lower bounds for the index of minimal spheres as well as the optimal
isoperimetric inequality for Laplacian eigenvalues on the projective plane.

####
**
November 12, 2019 Tue, 3:30-4:30PM, Eckhart 202
**

Tristan Collins, MIT

**
Some results in Strominger-Yau-Zaslow Mirror Symmetry:
**
Mirror symmetry originally arose as a mysterious duality between Calabi-Yau threefolds, interchanging complex
and symplectic structures. This duality has since expanded to include a much broader collection of objects,
including Fano manifolds, and Landau-Ginzburg models. Two fundamental themes in mirror symmetry are (1) the
existence of special Lagrangian fibrations, as conjectured by Strominger-Yau-Zaslow and (2) the correspondence
between "stable" objects as predicted in work of Thomas-Yau, and Douglas. Here, stable objects are meant to be
special Lagrangian manifolds on the symplectic side, and holomorphic bundles with canonical metrics, on the
complex side. I will report on recent results in both of these directions. This talk with discuss joint works
with A. Jacob, Y.-S. Lin, and S.-T. Yau.

####
**
November 19, 2019 Tue, 3:30-4:30PM, Eckhart 202
**

Yang Li, IAS

**
Taub-NUT and Ooguri-Vafa: from 2D to 3D:
**
Taub-NUT and Ooguri-Vafa metrics are S^1 invariant Calabi-Yau metrics in complex dimension 2 constructed via the
Gibbons-Hawking ansatz. They feature prominently in the complex 2D case of the SYZ conjecture. After reviewing
the basics we explain how a number of first principles dictate their construction. This insight enables us to
generalize the construction to complex dimension 3, which is expected to be relevant for the SYZ conjecture.

####
**
November 26, 2019 Tue, 3:30-4:30PM, Eckhart 202
**

Romain Petrides, IMJ, Paris Diderot University

**
Critical metrics for Laplace eigenvalues on Riemannian surfaces:
**
We investigate the general link between the critical unit area metrics for eigenvalues of the Laplace operator
on closed surfaces, and minimal immersions of these surfaces by eigenfunctions. We will discuss the existence or
non existence of such objects by variational methods.

####
**
December 3, 2019 Tue, 3:30-4:30PM, Eckhart 202
**

Demetre Kazaras, Stony Brook

**
Desingularizing 4-manifolds with positive scalar curvature:
**
We study 4-manifolds of positive scalar curvature (psc) with severe metric singularities along points and
embedded circles, establishing a desingularization process. To carry this out, we show that the bordism group of
closed 3-manifolds with psc metrics is trivial, using scalar-flat K{\"a}hler ALE surfaces recently discovered by
Lock-Viaclovsky. This allows us to prove a non-existence result for singular psc metrics on enlargeable
4-manifolds, partially confirming a conjecture of Schoen. We will also mention a new lower bound for the mass of
3d asymptotically flat manifolds with nonnegative scalar curvature in joint work with Bray, Khuri, and Stern.

### SPRING 2019

####
**
April 16, 2019 Tue, 3:00-4:00PM, Eckhart 202
**

Lucas Ambrozio, IAS

**
Comparing the total volume of a closed Riemannian manifold and the geometric invariants related to its
minimal hypersurfaces:
**
The variational methods used to find closed embedded minimal hypersurfaces in a closed Riemannian manifold allow
one to define several notions of "systole" and "width", which can be regarded as functionals on the space of
Riemannian metrics. In this talk, we will be interested in the properties of some of these functionals on the
space of unit volume metrics, focusing on the case of three dimensional spheres and real projective spaces. In
particular, we will look for (sharp) upper bounds on certain subsets of metrics and describe necessary
conditions satisfied by local maxima. This is joint work with Rafael Montezuma.

####
**
April 23, 2019 Tue, 3:00-4:00PM, Eckhart 202
**

Nicolau Aiex, UBC

**
TBA:
**
TBA

####
**
April 23, 2019 Tue, 4:00-5:00PM, Eckhart 202
**

Davi Maximo, UPenn

**
On the topology and index of minimal surfaces:
**
For an immersed minimal surface in R^3, we show that there exists a lower bound on its Morse index that depends
on the genus and number of ends, counting multiplicity. This improves, in several ways, an estimate we
previously obtained bounding the genus and number of ends by the index. Our new estimate resolves several
conjectures made by J. Choe and D. Hoffman concerning the classification of low-index minimal surfaces: we show
that there are no complete two-sided immersed minimal surfaces in R^3 of index two, complete embedded minimal
surface with index three, or complete one-sided minimal immersion with index one. This is joint work with Otis
Chodosh.

####
**
April 30, 2019 Tue, 3:00-4:00PM, Eckhart 202
**

Alessandro Carlotto, IAS/ETH-Zurich

**
Constrained deformations of positive scalar curvature metrics:
**
I will present a series of results concerning the interplay between two different curvature conditions, in the
special case when these are given by pointwise inequalities on the scalar curvature of a manifold, and the mean
curvature of its boundary. Such results lie at two conceptual levels: on the one hand at the level of
compatibility (i.e. is it possible to simultaneously satisfy the bounds, and what are the resulting topological
implications), on the other hand at the level of moduli space structure (i.e. what can one say about the
homotopy type of the associated space of metrics, when not empty, quotiented by the diffeomorphism group of the
background manifold). In particular, we give a complete topological characterization of those compact
3-manifolds that support Riemannian metrics of positive scalar curvature and mean-convex boundary and, in any
such case, we prove that the associated moduli space of metrics is path-connected. Furthermore, we show how our
methods can be refined so to construct continuous paths of positive scalar curvature metrics with minimal
boundary, and to obtain analogous conclusions in that context as well. Our work relies on a combination of
earlier fundamental contributions by Gromov-Lawson and Schoen-Yau, on the smoothing procedure designed by Miao,
and on the interplay of Perelman's Ricci flow with surgery and conformal deformation techniques introduced by
CodÃ¡ Marques in dealing with the closed case. This lecture is based on joint work with Chao Li (Princeton
University).

####
**
May 7, 2019 Tue, 3:00-4:00PM, Eckhart 202
**

Harrison Pugh, Notre Dame

**
TBA:
**
TBA

####
**
May 28, 2019 Tue, 3:00-4:00PM, Eckhart 202
**

Ben Sharp, University of Leeds

**
Global estimates for harmonic maps from surfaces:
**
A celebrated theorem of F. HÃ©lein guarantees that a weakly harmonic map from a two-dimensional domain is always
smooth. The proof is of a local nature and assumes that the Dirichlet energy is sufficiently small; under this
condition it is possible to re-write the harmonic map equation using a suitably chosen frame which uncovers
non-linearities with more favourable regularity properties (so-called div-curl or Wente structures). We will
prove a global estimate for harmonic maps without assuming a small energy bound, utilising a powerful theory
introduced by T. RiviÃ¨re. Along the way we will highlight the relevance of Wente-type estimates in neighbouring
areas of geometric analysis, and hint as to why the analogous higher-dimensional global estimate remains a
challenging open problem. This is a joint work with Tobias Lamm.

### WINTER 2019

####
**
January 15, 2019 Tue, 3:00-4:00PM, Eckhart 202
**

Henrik Matthiesen, The University of Chicago

**
The systole of large genus minimal surfaces in positive Ricci curvature:
**
We prove that the systole (or more generally, any $k$-th homology systole) of a minimal surface in an ambient
three manifold of positive Ricci curvature tends to zero as the genus of the minimal surfaces becomes unbounded.
This is joint work with Anna Siffert.

####
**
January 24, 2019 Thursday, 2:00-3:00PM, Eckhart 202
**

Xin Zhou, UCSB

**
Multiplicity One Conjecture in Min-max theory:
**
I will present a recent proof of the Multiplicity One Conjecture in Min-max theory. This conjecture was raised
by Marques and Neves. It says that in a closed manifold of dimension between 3 and 7 with a bumpy metric, the
min-max minimal hypersurfaces associated with the volume spectrum introduced by Gromov, Guth, Marques-Neves are
all two-sided and have multiplicity one. As direct corollaries, it implies the generalized Yau's conjecture for
such manifolds with positive Ricci curvature, which says that there exist infinitely many pairwise non-isometric
minimal hypersurfaces, and the Weighted Morse Index Bound Conjecture by Marques and Neves.

####
**
January 30, 2019 Wed, 3:00-4:00PM, Eckhart 202
**

Tristan Riviere, ETH

**
The Cost of the Sphere Eversion and the 16\pi Conjecture:
**
How much does it cost...to knot a closed simple curve ? To cover the sphere twice ? to realize such or such
homotopy class ? ...etc. All these questions consisting of assigning a "canonical" number and possibly an
optimal "shape" to a given topological operation are known to be mathematically very rich and to bring together
notions and techniques from topology, geometry and analysis. In this talk we will concentrate on the operation
consisting of turning inside out the 2 sphere in the 3 dimensional space. Since Smale's proof in 1959 of the
existence of such an operation the search for effective realizations of such eversions has triggered a lot of
fascination and works in the math community. The absence in nature of matter that can interpenetrate and the
quasi impossibility, up to the advent of virtual imaging, to experience this deformation is maybe the reason for
the difficulty to develop an intuitive approach on the problem. We will present the optimization of Sophie
Germain conformally invariant elastic energy for the eversion. Our efforts will finally bring us to consider
more closely an integer number together with a mysterious minimal surface.

####
**
Feb 12, 2019 Tue, 3:00-4:00PM, Eckhart 202
**

Vanderson Lima, UFRGS

**
Stable Minimal Surfaces and the topology of 3-Manifolds:
**
Meeks, Perez and Ros conjectured that a closed Riemannian 3-manifold which does not contain any closed embedded
stable minimal surface must be diffeomorphic to a quotient of the 3-sphere. In this talk we show a counter
example for this conjecture. We also discuss how to correct the conjecture so that it holds true.

####
**
Feb 19, 2019 Tue, 3:00-4:00PM, Eckhart 202
**

Casey Kelleher, Princeton University

**
INDEX-ENERGY ESTIMATES FOR YANG-MILLS CONNECTIONS AND EINSTEIN METRICS:
**
We prove a conformally invariant estimate for the index of Schrodinger operators acting on vector bundles over
four-manifolds, related to the classical Cwikel-Lieb-Rozenblum estimate. Applied to Yang-Mills connections we
obtain a bound for the index in terms of its energy which is conformally invariant, and captures the sharp
growth rate. Furthermore we derive an index estimate for Einstein metrics in terms of the topology and the
Einstein-Hilbert energy. Lastly we derive conformally invariant estimates for the Betti numbers of an oriented
four-manifold with positive scalar curvature.

####
**
Feb 26, 2019 Tue, 3:00-4:00PM, Eckhart 202
**

Giuseppe Tinaglia, King's College

**
The geometry of constant mean curvature surfaces in Euclidean space:
**
In this talk I will begin by reviewing classical geometric properties of constant mean curvature surfaces, H>0,
in R^3. I will then talk about several more recent results for surfaces embedded in R^3 with constant mean
curvature, such as curvature and radius estimates for simply-connected surfaces embedded in R^3 with constant
mean curvature. Finally I will show applications of such estimates including a characterisation of the round
sphere as the only simply-connected surface embedded in R^3 with constant mean curvature and area estimates for
compact surfaces embedded in a flat torus with constant mean curvature and finite genus. This is joint work with
Meeks.

####
**
Feb 26, 2019 Tue, 3:00-4:00PM, Eckhart 202
**

Gregory Chambers, Rice University

**
Geometric stability of the Coulomb energy:
**
Suppose that $A$ is a measurable subset of $\mathbb{R}^3$ of finite measure. The Coulomb energy of $A$ is the
double integral over $A$ of $1/|x-y|$, and is maximized when $A$ is a ball. If the Coulomb energy is close to
maximal, then is $A$ geometrically close to a ball? We will answer this question, and will compare it to the
quantitative isoperimetric inequality. We will also discuss the analogous situation in higher dimensions. This
is joint work with Almut Burchard

####
**
Mar 5, 2019 Tue, 3:00-4:00PM, Eckhart 202
**

Yaiza Canzani, UNC

**
Understanding the growth of Laplace eigenfunctions:
**
In this talk we will discuss a new approach to understanding eigenfunction concentration. We characterize the
features that cause an eigenfunction to saturate the standard supremum bounds in terms of the distribution of
L^2 mass along geodesic tubes emanating from a point. We also show that the phenomena behind extreme supremum
norm growth is identical to that underlying extreme growth of eigenfunctions when averaged along submanifolds.
Using the description of concentration, we obtain quantitative improvements on the known bounds in a wide
variety of settings.

####
**
Mar 12, 2019 Tue, 3:00-4:00PM, Eckhart 202
**

Yu Wang, Northwestern University

**
Sharp estimate of global Coulomb gauge in dimension 4:
**
Consider a principle SU(2)-bundle P over a compact 4-manifold M and a W^{1,2}-connection A of P satisfying
\|F_A\|_{L^2(M)}\le \Lambda. Our main result is the existence of a global section \sigma: M\to P with
controllably many singularities such that the connection form \sigma^*A satisfies the Coulomb equation
d^*(\sigma^*A)=0 and moreover admits a sharp estimate. In this talk, we first recall some preliminaries and then
outlines the proof. If time allows we shall elaborate on the ideas to overcome the main difficulties in this
problem, which include an epsilon-regularity theorem for the Coulomb gauge, an annular-bubble regions
decomposition for the curvature, and studying the singularity behavior of the Coulomb gauge on each annular and
bubble region.

####
**
Mar 19, 2019 Tue, 3:00-4:00PM, Eckhart 202
**

Rayssa Caju,

**
ASYMPTOTIC BEHAVIOUR OF SOLUTIONS FOR A COUPLED ELLIPTIC SYSTEM IN THE PUNCTURED BALL:
**
Our main goal is to study the asymptotic behavior near an isolated singularity of local solutions for certain
strongly coupled critical elliptic systems that, from the viewpoint of conformal geometry, are pure extensions
of Yamabe-type equations. There has been considerable interest in recent years in proving compactness results
for this type of systems since such type of problems provides a natural background for the interplay between
geometry and asymptotic analysis. We prove a sharp result on the removability of the isolated singularity for
all components of the solutions when the dimension is less than or equal to five and minus the potential of the
operator in the system is cooperative.

### SPRING 2018

####
**
April 3, 2018 Tue, 3:45-4:45PM, Eckhart 207
**

Lucas Ambrosio, University of Warwick

**
Sequences of minimal surfaces with bounded index in three-manifolds:
**
We explore a few consequences of the bubbling analysis developed by Buzano and Sharp, giving a detailed
description of how a sequence of closed embedded minimal surfaces of bounded index in a three-manifold
degenerates as we pass to a "converging" subsequence. In particular, a few new compactness result are obtained.
This is a joint work with Reto Buzano (QMUL), A. Carlotto (ETH) and B. Sharp (Leeds).

####
**
April 10, 2018 Tue, 3:45-4:45PM, Eckhart 207
**

Costante Bellettini, University College of London

**
Stable constant-mean-curvature hypersurfaces: regularity and compactness:
**
This talk describes a recent joint work of the speaker with N. Wickramasekera (Cambridge). The work develops a
regularity theory, with an associated compactness theorem, for weakly defined hypersurfaces (codimension 1
integral varifolds) of a smooth Riemannian manifold that are stationary and stable on their regular parts for
volume preserving ambient deformations. The main regularity theorem gives two structural conditions on such a
hypersurface that imply that, away from a set of codimension 7 or higher, the hypersurface is locally either a
single smoothly embedded disk or precisely two smoothly embedded disks intersecting tangentially. Easy examples
show that neither structural hypothesis can be relaxed. An important special case is when the varifold
corresponds to the boundary of a Caccioppoli set, in which case the structural conditions can be considerably
weakened. An "effective version" of the compactness theorem has been (a posteriori) established in collaboration
with O. Chodosh and N. Wickramasekera.

####
**
April 17, 2018 Tue, 3:00-4:00PM, Eckhart 202
**

Daniel Agress, University of California Irvine

**
Existence results for the nonlinear Hodge minimal surface energy:
**
The nonlinear Hodge minimal surface energy, first studied by Sibner and Sibner in the 1970's, has applications
to minimal surfaces, bounded variation functions, and the Born Infeld theory of electromagnetism. In this talk,
we will prove an existence and nonexistence result for minimizers of the energy. In particular, we show that for
a compact Riemannian manifold and cohomology class $[\alpha] \in H^k(M)$, minimizers always exist when $k=1$,
but counterexamples exist when $k>1$. We will also describe the how the energy can be viewed as a regularization
of the BV energy.

####
**
April 24, 2018 Tue, 3:45-4:45PM, Eckhart 207
**

Mikhail Karpukhin, McGill University

**
Laplace eigenvalues and minimal surfaces in spheres:
**
We will give an overview of some recent estimates for Laplace eigenvalues on Riemannian surfaces. In particular,
we will discuss the connection of optimal isoperimetric inequalities with minimal surfaces and harmonic maps.
Finally, this connection will be used in order to prove the sharp upper bound for all Laplace eigenvalues on the
two-dimensional sphere. The talk is based on a joint work with N. Nadirashvili, A. Penskoi and I. Polterovich.

####
**
April 30, 2018 Mon, 3:45-4:45PM, Eckhart 207 Please note the unusual day
**

Peter Smillie, Harvard University

**
Entire spacelike surfaces of constant curvature in Minkowski 3-space:
**
We prove that every regular domain in Minkowski 3-space which is not a wedge contains a unique entire spacelike
surface with constant intrinsic curvature equal to -1. This completes the classification of such surfaces in
terms of their domains of dependence, for which partial results were obtained by Li, Guan-Jian-Schoen, and
Bonsante-Seppi. Using this result, we obtain an analogous classification of entire spacelike surfaces with
constant mean curvature (CMC). We'll apply these ideas to the Minkowski problem of prescribed curvature and to
the construction CMC times in 2+1 relativity, and we'll see what we can say about the problem of deciding when
the induced hyperbolic metric on an entire surface is complete. Everything is joint with Francesco Bonsante and
Andrea Seppi.

####
**
May 8, 2018 Tue, 3:45-4:45PM, Eckhart 207
**

Rafael Montezuma, Princeton University

**
A mountain pass theorem for minimal hypersurfaces with fixed boundary:
**
In this talk, we will be concerned with the existence of a third embedded minimal hypersurface, of mountain-pass
type, spanning a closed submanifold B contained in the boundary of a compact Riemannian manifold with convex
boundary, when it is known a priori the existence of two strictly stable minimal hypersurfaces that bound B. In
order to do so, we develop min-max methods similar to those of the recent work of De Lellis and Ramic, adapted
to the discrete setting of Almgren and Pitts.

####
**
May 15, 2018 Tue, 3:45-4:45PM, Eckhart 207
**

Pei-Ken Hung, Columbia University

**
The smoothing time of convex inverse mean curvature flows:
**
By using the local estimate recently proved by B. Choi and P. Daskalopoulos, we show that the smoothing time of
convex inverse mean curvature flows is given by the smallest area of the initial tangent cone. As a corollary,
convex inverse mean curvature flows on the sphere become smooth before the extinction time unless the
corresponding cone splits a line. This is ongoing joint work with B. Choi.

####
**
May 22, 2018 Tue, 3:45-4:45PM, Eckhart 207
**

Bing Wang, University of Wisconsin - Madison

**
The extension problem of the mean curvature flow in R^3:
**
We show that the mean curvature blows up at the first finite singular time for a closed smooth embedded mean
curvature flow in R^3. This is a joint work with H.Z. Li.

####
**
June 5, 2018 Tue, 3:00-4:00PM, Eckhart 207 Please note we will have two talks on the same day
**

Xinliang An, U of Toronto

**
How to Make a Black Hole:
**
Black holes are predicted by Einstein's theory of general relativity, and now we have ample observational
evidence for their existence. However theoretically there are many unanswered questions about how black holes
come into being. In this talk, with tools from hyperbolic PDE, quasilinear elliptic equations, geometric
analysis and dynamical systems, we will prove that, through a nonlinear focusing effect, initially low-amplitude
and diffused gravitational waves can give birth to a black hole region in our universe. This result extends the
1965 Penrose's singularity theorem and it also proves a conjecture of Ashtekar on black-hole thermodynamics.
Open problems and new directions will also be discussed.

####
**
June 5, 2018 Tue, 4:15-5:15PM, Eckhart 207 Please note we will have two talks on the same day
**

Allen Brian, USMA West Point

**
Using IMCF to show stability of the Positive Mass Theorem and Riemannian Penrose Inequality:
**
In this talk we will discuss the stability of the Positive Mass Theorem (PMT) and Riemannian Penrose Inequality
(RPI) for a sequence of compact regions of asymptotically flat manifolds, $U_T \subset M^3$, which are foliated
by a uniformly controlled solution to Inverse Mean Curvature Flow (IMCF). The PMT result says that if the
Hawking Mass of the outermost boundary of the regions, $U_T$, is approaching zero than the metric on $U_T$ is
converging to the Euclidean metric in $L^2$. A similar result for the RPI will also be discussed where the
metric on $U_T$ converges to the Schwarzschild metric in $L^2$. Key estimates will be introduced and ideas for
how to extend this work will be explored.

### WINTER 2018

####
**
Jan 9, 2018 Tue, 3:45-4:45PM, Eckhart 217
**

Nicolau Aiex, University of British Columbia

**
The space of min-max hypersurfaces:
**
We use Lusternik-Schnirelmann Theory to study the topology of the space of closed embedded minimal hypersurfaces
on a manifold of dimension between 3 and 7 and positive Ricci curvature. Combined with the works of
Marques-Neves we can also obtain some information on the geometry of the minimal hypersurfaces they found.

####
**
Jan 16, 2018 Tue, 3:45-4:45PM, Eckhart 217
**

Robin Neumayer, Northwestern University

**
The Cheeger constant of a Jordan domain without necks:**
In 1970, Cheeger established lower bounds on the first eigenvalue of the Laplacian on compact Riemannian
manifolds in terms of a certain isoperimetric problem. The analogous problem on domains of Euclidean space has
generated much interest in recent years, due in part to its connections to capillarity theory, image processing,
and landslide modeling. In this talk, based on joint work with Leonardi and Saracco, we give an explicit
characterization of minimizers in this isoperimetric problem for a very general class of planar domains.

####
**
Jan 30, 2018 Tue, 3:45-4:45PM, Eckhart 217
**

Kei Irie, Kyoto University

**
Denseness of closed geodesics on surfaces with generic Riemannian metrics:
**
We prove that, on a closed surface with a $C^\infty$-generic Riemannian metric, the union of nonconstant closed
geodesics is dense. This result follows from a more general result about periodic orbits of Reeb dynamics on
contact three-manifolds. The proof uses embedded contact homology (ECH), a version of Floer homology definedfor
contact three-manifolds, which was introduced by Hutchings. In particular, the key ingredient is the ``Weyl
law'' for ECH spectral numbers, which was proved by Cristofaro-Gardiner, Hutchings, and Ramos. We also discuss a
denseness of minimal hypersurfaces for genericmetrics (joint work with Marques and Neves), which was obtained by
applying a similar idea to the Weyl law for volume spectrum, which was proved by Liokumovich, Marques, and
Neves.

####
**
Feb 6, 2018 Tue, 3:45-4:45PM, Eckhart 217
**

Mathew Langford, The University of Tennessee

**
Ancient solutions of the mean curvature flow:**
I will present a survey of existence and rigidity results for ancient solutions of mean curvature flow. In
particular, I will describe recent work (with Theodora Bourni and Giuseppe Tinaglia) on the existence and
uniqueness of rotationally symmetric ancient solutions which lie in a slab. Time permitting, we will finish by
describing some interesting open problems.

####
**
Feb 12, 2018 Mon, 3:45-4:45PM, Eckhart 202 Please note the unusual day
**

Theodora Bourni, The University of Tennessee

**
Ancient Pancakes:
**
We show that, up to rigid motions, there is a unique compact, convex, rotationally symmetric, ancient solution
of mean curvature flow that lies in a slab of width $\pi$ and in no smaller slab. This is joint work with Mat
Langford and Giuseppe Tinaglia

####
**
Feb 20, 2018 Tue, 3:45-4:45PM, Eckhart 217
**

Zahra Sinaei, Northwestern University

**
TBA:
**
In this talk, I discuss partial regularity of stationary solutions and minimizers u from a set \Omega\subset
\R^n to a Riemannian manifold N, for the functional \int_\Omega F(x,u,|\nabla u|^2) dx. The integrand F is
convex and satisfies some ellipticity, boundedness and integrability assumptions. Using the idea of quantitative
stratification I show that the k-th strata of the singular set of such solutions are k-rectifiable.

####
**
Feb 27, 2018 Tue, 3:45-4:45PM, Eckhart 217
**

Jesse Madnick, Stanford University

**
TBA:
**
TBA

####
**
Mar 6, 2018 Tue, 3:45-4:45PM, Eckhart 217
**

Panagiotis Gianniotis, University of Toronto

**
The bounded diameter conjecture for 2-convex mean curvature flow:**
In this talk I address the bounded diameter conjecture for the mean curvature flow of smooth 2-convex
hypersurfaces in $R^{n+1}$. In joint work with Robert Haslhofer, we prove that the intrinsic diameter of the
evolving hypersurfaces is controlled, up to the first singular time, in terms of geometric information of the
initial hypersurface. Moreover, this diameter estimate leads to sharp $L^{nâ1}$ estimates for the curvature at
each time. Our estimates extend to mean curvature flow with surgery, which allows us to obtain the optimal
$L^{nâ1}$ estimate for any level set flow starting from a smooth 2-convex hypersurface. This improves the
$L^{nâ1â\varepsilon}$ curvature estimate that was previously established in work of Head and
Cheeger-Haslhofer-Naber.

### FALL 2017

####
**
Oct 3, 2017 Tue, 3:30-4:30PM, Eckhart 358
**

Antoine Song, Princeton University

**
Local min-max surfaces and existence of minimal Heegaard splittings:
**
Let M be a closed oriented 3-manifold not diffeomorphic to the 3-sphere, and suppose that there is a strongly
irreducible Heegaard splitting H. Previously, Rubinstein announced that either there is a minimal surface of
index at most one isotopic to H or there is a non-orientable minimal surface such that the double cover with a
vertical handle attached is isotopic to H. He sketched a natural outline of a proof using min-max, however some
steps are non-trivially incomplete and we will explain how to justify them. The key point is a version of
min-max theory producing interior minimal surfaces when the ambient manifold has minimal boundary. Some
corollaries of the theorem include the existence in any RP^3 of either a minimal torus or a minimal projective
plane with stable universal cover. Several consequences for metric with positive scalar curvature are also
derived.

####
**
Oct 10, 2017 Tue, 3:30-4:30PM, Eckhart 358
**

Renato Bettiol, University of Pennsylvania

**
Non-uniqueness of conformal metrics with constant Q-curvature:
**
The problem of finding (complete) metrics with constant Q-curvature in a prescribed conformal class is a famous
fourth-order cousin of the Yamabe problem. In this talk, I will provide some background on Q-curvature and
discuss how several non-uniqueness results for the Yamabe problem can be transplanted to this context. However,
special emphasis will be given to multiplicity phenomena for constant Q-curvature that have no analogues for the
Yamabe problem, confirming expectations raised by the lack of a maximum principle.

####
**
Oct 17, 2017 Tue, 3:30-4:30PM, Eckhart 358
**

Otis Chodosh, Princeton University

**
Minimal surfaces in asymptotically flat 3-manifolds:
**
The study of minimal surfaces in asymptotically flat 3-manifolds goes back to the proof of the positive mass
theorem by Schoen and Yau. I'll explain a rigidity theorem (joint with M. Eichmair) and an existence theorem
(joint with D. Ketover) concerning such surfaces.

####
**
Oct 24, 2017 Tue, 3:30-4:30PM, Eckhart 358
**

Chao Li, Stanford University

**
A polyhedron comparison theorem in 3-manifolds with positive scalar curvature:
**
We establish a comparison theorem for polyhedrons in 3-manifolds with nonnegative scalar curvature, answering
affirmatively the dihedral rigidity conjecture by Gromov. For a large collections of polyhedrons with interior
non-negative scalar curvature and mean convex faces, we prove the dihedral angles along its edges cannot be
everywhere less than those of the corresponding Euclidean model. We also establish the rigidity case.

####
**
Oct 31, 2017 Tue, 3:30-4:30PM, Eckhart 358
**

Robert Haslhofer, University of Toronto

**
Minimal two-spheres in three-spheres:
**
We prove that any manifold diffeomorphic to S^3 and endowed with a generic metric contains at least two embedded
minimal two-spheres. The existence of at least one minimal two-sphere was obtained by Simon-Smith in 1983. Our
approach combines ideas from min-max theory and mean curvature flow. We also establish the existence of smooth
mean convex foliations in three-manifolds. Finally, we apply our methods to solve a problem posed by S.T. Yau in
1987, and to show that the assumptions in the multiplicity one conjecture and the equidistribution of widths
conjecture are in a certain sense sharp. This is joint work with Dan Ketover.

####
**
Nov 8, 2017 Wed, 2:00-3:00PM, Eckhart 358 There will be two seminars on this date
**

Hannah Alpert, The Ohio State University

**
Morse broken trajectories and hyperbolic volume:
**
A large family of theorems all state that if a space is topologically complex, then the functions on that space
must express that complexity, for instance by having many singularities. For the theorem in this talk, our
preferred measure of topological complexity is the hyperbolic volume of a closed manifold admitting a hyperbolic
metric (or more generally, the Gromov simplicial volume of any space). A Morse function on a manifold with large
hyperbolic volume may still not have many critical points, but we show that there must be many flow lines
connecting those few critical points. Specifically, given a closed n-dimensional manifold and a Morse-Smale
function, the number of n-part broken trajectories is at least the Gromov simplicial volume. To prove this we
adapt lemmas of Gromov that bound the simplicial volume of a stratified space in terms of the complexity of the
stratification.

####
**
Nov 8, 2017 Wed, 3:00-4:00PM, Eckhart 358 There will be two seminars on this date
**

Jose Maria Espinar, IMPA - Brazil

**
Characterization of f-extremal disks:
**
In this talk we show uniqueness for overdetermined elliptic problems defined on topological disks with regular
boundary, i.e., positive solutions $u$ to $\Delta u + f(u)=0$ in $\Omega \subset (M^2,g)$ so that $u = 0$ and
$\frac{\partial u}{\partial \vec\eta} = cte $ along $\partial \Omega$, $\vec\eta$ the unit outward normal along
$\partial\Omega$ under the assumption of the existence of a candidate family. In particular, this gives a
positive answer to the Schiffer conjecture for the first Dirichlet eigenvalue and classifies simply-connected
harmonic domains, also called {\it Serrin Problem}, in $\mathbb S ^2$. This is a joint work with L. Mazet.

####
**
Nov 15, 2017 Wed, 2:00-3:00PM, Eckhart 358
**

GÃ¡bor SzÃ©kelyhidi, University of Notre Dame

**
New Calabi-Yau metrics on C^n:
**
I will discuss the construction of Calabi-Yau metrics on C^n with maximal volume growth, with singular tangent
cones at infinity. These generalize recent examples constructed independently by Yang Li and Conlon-Rochon

####
**
Nov 22, 2017 Wed, 2:00-3:00PM, Eckhart 358
**

Fritz Hiesmayr, University of Cambridge

**
Index and spectrum of minimal hypersurfaces arising from the Allen-Cahn construction:
**
The Allen-Cahn construction is a method for constructing minimal surfaces of codimension 1 in closed manifolds.
In this approach, minimal hypersurfaces arise as the weak limits of level sets of critical points of the
Allen-Cahn energy functional. This talk will relate the variational properties of the Allen-Cahn energy to those
of the area functional on the surface arising in the limit, under the assumption that the limit surface is
two-sided. In this case, bounds for the Morse indices of the critical points lead to a bound for the Morse index
of the limit minimal surface. As a corollary, minimal hypersurfaces arising from an Allen-Cahn p-parameter
min-max construction have index at most p. An analogous argument also establishes a lower bound for the spectrum
of the Jacobi operator of the limit surface.

### SPRING 2017

####
**
Apr 4, 2017 Tue, 3:00-4:00PM, Eckhart 358
**

Shmuel Weinberger, The University of Chicago

**
Persistent Homology and Gromov's theorem on closed contractible geodesics:
**
Gromov showed that on any closed Riemannian manifold whose fundamental group has unsolvable word problem, there
are infinitely many closed nullhomotopic geodesics (of index 0). I will explain this theorem from the point of
Persistent Homology of the free loop space, and then give some refinements (e.g. to less exotic fundamental
groups) and extensions (to some other functionals other than energy on loops).

####
**
Apr 11, 2017 Tue, 3:00-4:00PM, Eckhart 358
**

David Wiygul, UC Irvine

**
The Bartnik-Bray outer mass of small spheres:
**
In 1989 Robert Bartnik proposed a definition of quasilocal mass in general relativity. The Bartnik mass is known
to enjoy several attractive properties but is not straightforward to evaluate. I will talk about a first-order
estimate for a natural modification of Bartnik's definition applied to small perturbations of spheres in
Euclidean space. In particular I will describe an application to the small-sphere limit in time-symmetric
slices.

####
**
Apr 18, 2017 Tue, 3:00-4:00PM, Eckhart 358
**

Nick Edelen , MIT

**
Quantitative Reifenberg for Measures:
**
In joint work with Aaron Naber and Daniele Valtorta, we demonstrate a quantitative structure theorem for
measures in R^n under assumptions on the Jones \beta-numbers, which measure how close the support is to being
contained in a subspace. Measures with this property have arisen in several interesting scenarios: in obtaining
packing estimates on and rectifiability of the singular set of minimal surfaces; in characterizing
L2-boundedness of Calderon-Zygmund operators; and as an "annalist's" formulation of the travelling salesman
problem.

####
**
Apr 25, 2017 Tue, 3:00-4:00PM, Eckhart 358
**

Dan Ketover , Princeton University

**
Free boundary minimal surfaces of unbounded genus:
**
Free boundary minimal surfaces are natural variational objects that have been studied since the 40s. In spite of
this, very few explicit examples in the simplest case of the round three ball are known. I will describe how
variational methods can be used to construct new examples with unbounded genus resembling a desingularization of
the critical catenoid and flat disk. I will also give a new variational interpretation of the previously known
examples.

####
**
May 02, 2017 Tue, 3:00-4:00PM, Eckhart 358
**

Marco Radeschi , University of Notre Dame

**
Minimal hypersurfaces in compact symmetric spaces:
**
A conjecture of Marques-Neves-Schoen says that for every embedded minimal hypersurface M in a manifold of
positive Ricci curvature, the first Betti number of M is bounded above linearly by the index of M. We will show
that for every compact symmetric space this result holds, up to replacing the index of M with its extended
index. Moreover, for special symmetric spaces, the actual conjecture holds for all metrics in a neighbourhood of
the canonical one. These results are a joint work with R. Mendes.

####
**
May 16, 2017 Tue, 3:00-4:00PM, Eckhart 358
**

Pierre Albin , University of Illinois at Urbana-Champaign

**
Analytic torsion of manifolds with fibered cusps:
**
Analytic torsion is a spectral invariant of the Hodge Laplacian of a manifold with a flat connection. On a
closed manifold it is equal to a topological invariant known as Reidemeister torsion. I will describe joint work
with FrÃ©dÃ©ric Rochon and David Sher establishing a topological expression for the analytic torsion of a
manifold with fibered cusp ends (such as a locally symmetric space of rank one). We establish our result by
controlling the behavior of the spectrum along a degenerating class of Riemannian metrics.

####
**
May 23, 2017 Tue, 3:00-4:00PM, Eckhart 358 There will be two seminars on this date
**

Jacob Bernstein , Johns Hopkins University

**
Surfaces of Low Entropy:
**
Following Colding and Minicozzi, we consider the entropy of (hyper)-surfaces in Euclidean space. This is a
numerical measure of the geometric complexity of the surface and is intimately tied to to the singularity
formation of the mean curvature flow. In this talk, I will discuss several results that show that closed
surfaces for which the entropy is small are simple in various senses. This is all joint work with L. Wang.

####
**
May 23, 2017 Tue, 4:30-5:30PM, Eckhart 358 There will be two seminars on this date
**

Pedro Gaspar , IMPA

**
Minimal hypersurfaces and the Allen-Cahn equation on closed manifolds:
**
Since the late 70s parallels between the theory of phase transitions and critical points of the area functional
have helped us to understand variational properties of certain semi-linear elliptic PDEs and spaces of
hypersurfaces which minimize the area in an appropriate sense. We will discuss some recent developments in this
direction which extend well-known analogies regarding minimizers to more general variational solutions. In
particular, borrowing ideas from the min-max theory of minimal hypersurfaces, we study the number of solutions
of the Allen-Cahn equation in a closed manifolds and solutions with least non-trivial energy.

####
**
May 30, 2017 Tue, 3:00-4:00PM, Eckhart 358
**

Or Hershkovits , Stanford University

**
Uniqueness of mean curvature flow with mean convex singularities:
**
Given a smooth compact hypersurface in Euclidean space, one can show that there exists a unique smooth evolution
starting from it, existing for some maximal time. But what happens after the flow becomes singular? There are
several notions through which one can describe weak evolutions past singularities, with various relationship
between them. One such notion is that of the level set flow. While the level set flow is almost by definition
unique, it has an undesirable phenomenon called fattening: Our "weak evolution" of n-dimensional hypersurfaces
may develop (and does develop in some cases) an interior in R^{n+1}. This fattening is, in many ways, the right
notion of non-uniqueness for weak mean curvature. As was alluded to above, fattening can not occur as long as
the flow is smooth. Thus it is reasonable to say that the source of fattening is singularities. Permitting
singularities, it is very easy to show that fattening does not occur if the initial hypersurface, and thus all
the evolved hypersurface, are mean convex. Thus, singularities encountered during mean convex mean curvature
flow should be of the kind that does not create singularities (i.e, the local structure of the singularities
should prevent fattening, without any global mean convexity assumption). To put differently, it is reasonable to
conjecture that: "An evolving surface cannot fatten unless it has a singularity with no spacetime neighborhood
in which the surface is mean convex". In this talk, we will phrase a concrete formulation of this conjecture,
and describe its proof. This is a joint work with Brian White.

####
**
June 05, 2017 Tue, 3:00-4:00PM, Eckhart 358
**

Xin Zhou , MIT

**
Min-max minimal hypersurfaces with free boundary:
**
I will present a joint work with Martin Li. Minimal surfaces with free boundary are natural critical points of
the area functional in compact smooth manifolds with boundary. In this talk, I will describe a general existence
theory for minimal surfaces with free boundary. In particular, I will show the existence of a smooth embedded
minimal hypersurface with free boundary in any compact smooth Euclidean domain. The minimal surfaces with free
boundary were constructed using the min-max method. Our result allows the min-max free boundary minimal
hypersurface to be improper; nonetheless the hypersurface is still regular.

####
**
June 29, 2017 Thu, 3:00-4:00PM, Eckhart 202
**

Henrik Matthiesen , Max Plank Institute for Mathematics Bonn

**
Existence of metrics maximizing the first eigenvalue on closed surfaces:
**
We show that on each closed surface of fixed topological type, orientable or non-orientable, there is a metric,
smooth away from finitely many conical singularities, that maximizes the first eigenvalue among all unit area
metrics. The key new ingredient are several monotonicity results relating the corresponding maximal eigenvalues.
This is joint work with Anna Siffert.

### WINTER 2017

####
**
Jan 10, 2017 Tue, 3:00-4:00PM, Eckhart 312
**

Gang Liu, Northwestern University

**
On some recent progress of Yau's uniformization conjecture:
**
Yau's uniformization conjecture states that a complete noncompact Kahler manifold with positive bisectional
curvature is biholomorphic to the complex Euclidean space. We shall discuss some recent progress via the
Gromov-Hausdorff convergence technique.

####
**
Jan 17, 2017 Tue, 3:00-4:00PM, Eckhart 312
**

John Ma, University of British Columbia

**
Compactness, finiteness properties of Lagrangian self-shrinkers in R^4 and Piecewise mean curvature flow:
**
In this talk, we discuss a compactness result on the space of compact Lagrangian self-shrinkers in R^4. When the
area is bounded above uniformly, we prove that the entropy for the Lagrangian self-shrinking tori can only take
finitely many values; this is done by deriving a Lojasiewicz-Simon type gradient inequality for the branched
conformal self-shrinking tori. Using the finiteness of entropy values, we construct a piecewise Lagrangian mean
curvature flow for Lagrangian immersed tori in R^4, along which the Lagrangian condition is preserved, area is
decreasing, and the type I singularities that are compact with a fixed area upper bound can be perturbed away in
finite steps. This is a Lagrangian version of the construction for embedded surfaces in R^3 by Colding and
Minicozzi. This is a joint work with Jingyi Chen.

####
**
Jan 24, 2017 Tue, 3:00-4:00PM, Ryerson 358
**

Laura Schaposnik, University of Illinois at Chicago

**
On the geometry of branes in the moduli space of Higgs bundles:
**
After giving a gentle introduction to Higgs bundles, their moduli space and the associated Hitchin fibration, I
shall describe how actions both on groups and on surfaces (anti-holomorphic or of finite groups) lead to
families of interesting subspaces of the moduli space of Higgs bundles (the so-called branes). Finally, we shall
look at correspondences between these branes that arise from Langlands duality, as well as from other relations
between Lie groups.

####
**
Jan 31, 2017 Tue, 3:00-4:00PM, Ryerson 358
**

Pei-Ken Hung, Columbia University

**
A Minkowski inequality for hypersurfaces in the Anti-deSitter-Schwarzschild manifold:
**
We prove a sharp inequality for hypersurfaces in the Anti-deSitter-Schwarzschild manifold. This inequality
generalizes the classical Minkowski inequality for surfaces in the three dimensional Euclidean space, and has a
natural interpretation in terms of the Penrose inequality for collapsing null shells of dust. The proof relies
on a monotonicity formula for inverse mean curvature flow, and uses a geometric inequality established by
Brendle.

####
**
Feb 7, 2017 Tue, 3:00-4:00PM, Ryerson 358
**

Lan-Hsuan Huang, University of Connecticut

**
Geometry of Static Asymptotically Flat 3-Manifolds:
**
A static potential on a 3-manifold reflects the symmetry of the spacetime development and is also closely
related to the scalar curvature deformation. We discuss how minimal surfaces of 3-manifold interact with the
static potential and, as an application, give a new proof to the rigidity of the Riemannian positive mass
theorem.

####
**
Feb 14, 2017 Tue, 3:00-4:00PM, Ryerson 358
**

Yu Li, University of Wisconsin-Madison

**
Ricci flow on asymptotically Euclidean manifolds:
**
In this talk, we prove that if an asymptotically Euclidean (AE) manifold with nonnegative scalar curvature has
long time existence of Ricci flow, it converges to the Euclidean space in the strong sense. By convergence, the
mass will drop to zero as time tends to infinity. Moreover, in three dimensional case, we use Ricci flow with
surgery to give an independent proof of positive mass theorem.

####
**
Feb 21, 2017 Tue, 3:00-4:00PM, Ryerson 358
**

Yannick Sire, Johns Hopkins University

**
Bounds on eigenvalues on Riemannian surfaces:
**
In a joint program with N. Nadirashvili, we developed some tools to prove existence of extremal metrics in
conformal classes for eigenvalues of the Laplace-Beltrami operator on surfaces. I will describe these results
and move to an application to the isoperimetric inequality of the third eigenvalue on the 2-sphere.

####
**
Feb 28, 2017 Tue, 3:00-4:00PM, Ryerson 358 There will be two seminars on this date
**

Henri Roesch, Duke University

**
Proof of a Null Penrose conjecture using a new Quasi-local Mass:
**
We define an explicit quasi-local mass functional which is non-decreasing along all foliations (satisfying a
convexity assumption) of null cones. We use this new functional to prove the null Penrose conjecture under
fairly generic conditions.

####
**
Feb 28, 2017 Tue, 4:00-5:00PM, Eckhart 206 There will be two seminars on this date - Note the different
room
**

Luca Spolaor, MIT

**
A direct approach to an epiperimetric inequality for Free-Boundary problems:
**
Using a direct approach, we prove a 2-dimensional epiperimetric inequality for the one-phase problem in the
scalar and vectorial cases and for the double-phase problem. From this we deduce the $C^{1,\alpha}$ regularity
of the free-boundary in the scalar one-phase and double-phase problems, and of the reduced free-boundary in the
vectorial case, without any restriction on the sign of the component functions. In this talk I will try to
explain the proof of the epiperimetric inequality in the scalar one-phase problem. This is joint work with
Bozhidar Velichkov.

####
**
March 7, 2017 Tue, 3:00-4:00PM, Ryerson 358
**

Victoria Sadovskaya, Department of Mathematics - Penn State

**
Boundedness, compactness, and invariant norms for operator-valued cocycles:
**
We consider group-valued cocycles over dynamical systems with hyperbolic behavior, such as hyperbolic
diffeomorphisms or subshifts of finite type. The cocycle A takes values in the group of invertible bounded
linear operators on a Banach space and is Holder continuous. We consider the periodic data of A, i.e. the set of
its return values along the periodic orbits in the base. We show that if the periodic data of A is bounded or
contained in a compact set, then so is the cocycle. Moreover, in the latter case the cocycle is isometric with
respect to a Holder continuous family of norms. This is joint work with B. Kalinin.

####
**
Mar 14, 2017 Tue, 3:00-4:00PM, Ryerson 358
**

Baris Coskunuser , Boston College

**
Embeddedness of the solutions to the H-Plateau Problem:
**
In this talk, we will give a generalization of Meeks and Yau's embeddedness result on the solutions of the
Plateau problem to the constant mean curvature disks. arXiv:1504.00661

### FALL 2016

####
**
Oct 11, 2016 Tue, 3:00-4:00PM, Eckhart 207
**

Lu Wang, Department of Mathematics - University of Wisconsin-Madison

**
Asymptotic structure of self-shrinkers:
**
We show that each end of a noncompact self-shrinker in Euclidean 3-space of finite topology is smoothly
asymptotic to a regular cone or a self-shrinking round cylinder.

####
**
Oct 18, 2016 Tue, 3:00-4:00PM, Eckhart 207
**

Spencer Becker-Kahn, Department of Mathematics - MIT

**
Zero Sets of Smooth Functions and p-Sweepouts:
**
It is well known that any closed set can occur as the zero set of a smooth function. But in many contexts in
analysis, one does not work with arbitrary smooth functions; one works with smooth functions that vanish to only
finite order. And the zero set of any such function must have some regularity and structure. With recourse to an
analogy made by Gromov and some recent conjectures of Marques and Neves, I will explain the application of some
general results about such zero sets to p-sweepouts (which are p-dimensional families of generalized
hypersurfaces that sweepout a smooth manifold in a certain sense). In the course of doing so, I will explain a
new result about the regularity of zero sets of smooth functions near a point of finite vanishing order (joint
work with Tom Beck and Boris Hanin).

####
**
Nov 8, 2016 Tue, 3:00-4:00PM, Eckhart 207 There will be two seminars on this date
**

Jonathan Zhu, Department of Mathematics - Harvard University

**
Entropy and self-shrinkers of the mean curvature flow:
**
The Colding-Minicozzi entropy is an important tool for understanding the mean curvature flow (MCF), and is a
measure of the complexity of a submanifold. Together with Ilmanen and White, they conjectured that the round
sphere minimises entropy amongst all closed hypersurfaces. We will review the basics of MCF and their theory of
generic MCF, then describe the resolution of the above conjecture, due to J. Bernstein and L. Wang for
dimensions up to six and recently claimed by the speaker for all remaining dimensions. A key ingredient in the
latter is the classification of entropy-stable self-shrinkers that may have a small singular set.

####
**
Nov 8, 2016 Tue, 4:00-5:00PM, Eckhart 207 There will be two seminars on this date
**

Greg Chambers, Department of Mathematics - The University of Chicago

**
Existence of minimal hypersurfaces in non-compact manifolds:
**
I prove that every complete non-compact manifold of finite volume admits a minimal hypersurface of finite area.
This work is in collaboration with Yevgeny Liokumovich.

####
**
Nov 15, 2016 Tue, 3:00-4:00PM, Eckhart 207
**

Peter McGrath, Brown University

**
New doubling constructions for minimal surfaces:
**
I will discuss recent work (with Nikolaos Kapouleas) on constructions of new embedded minimal surfaces using
singular perturbation 'gluing' methods. I will discuss at length doublings of the equatorial S^2 in S^3. In
contrast to earlier work of Kapouleas, the catenoidal bridges may be placed on arbitrarily many parallel circles
on the base S^2. This necessitates a more detailed understanding of the linearized problem and more exacting
estimates on associated linearized doubling solutions.

####
**
Nov 22, 2016 Tue, 3:00-4:00PM, Eckhart 207 There will be two seminars on this date
**

Franco Vargas Pallete, UC Berkeley

**
Renormalized volume:
**
The renormalized volume V_R is a finite quantity associated to certain hyperbolic 3-manifolds of infinite
volume. In this talk I'll discuss its definition and some properties for acylindrical manifolds, namely local
convexity and convergence under geometric limits. If time suffices I'll also discuss some applications and
further examples.

####
**
Nov 22, 2016 Tue, 4:00-5:00PM, Eckhart 207 There will be two seminars on this date
**

Christos Mantoulidis, Stanford University

**
Fill-ins, extensions, scalar curvature, and quasilocal mass:
**
There is a special relationship between the Jacobi and the ambient scalar curvature operator. I'll talk about an
extremal bending technique that exploits this relationship. It lets us compute the Bartnik mass of apparent
horizons and disprove a form of the Hoop conjecture due to G. Gibbons. Then, I'll talk about a derived
"cut-and-fill" technique that simplifies 3-manifolds of nonnegative scalar curvature. It is used in studying a
priori L^1 estimates for boundary mean curvature for compact initial data sets, and in generalizing Brown-York
mass. Parts of this talk reflect work done jointly with R. Schoen/P. Miao.

####
**
Nov 29, 2016 Tue, 3:00-4:00PM, Eckhart 207
**

Hung Tran, UC Irvine

**
Index Characterization for Free Boundary Minimal Surfaces:
**
A FBMS in the unit Euclidean ball is a critical point of the area functional among all surfaces with boundaries
in the unit sphere, the boundary of the ball. The Morse index gives the number of distinct admissible
deformations which decrease the area to second order. In this talk, we explain how to compute the index from
data of two simpler problems. The first one is the corresponding problem with fixed boundary condition; and the
second is associated with the Dirichlet-to-Neumann map for Jacobi fields. We also discuss applications to a
conjecture about FBMS with index 4.